Enter, I-B-VP-3-Ex1-v2, 2026, pp. 1-39
I-B-VP-3-Ex1 Massive second-order insertions, defect renormalization, and controlled massless limits
Ngo Cheung
18 October 2026
Abstract
We investigate the proposed extension of the massive conical vacuum-polarization programme to second order in a quartic coupling. Three corrections are indispensable: a fixed extension with a negative transverse bound state cannot reach zero mass through its above-threshold ground-state regime; a mass gap does not remove vertex-collision divergences; and the order-λ² two-point correction of real λφ⁴/4! theory is generally two-loop, not one-loop.
Two explicit constructions replace the unsupported general claim. First, we renormalize the repeated quadratic worldsheet insertion induced by the first-order apex contact. Its massive covariance, Wick square, and complete regular mixed stress reduce to convergent one-dimensional Macdonald integrals. Their compact-regular massless limits have an explicit error bound of order m²(1 + |log m|)³. At second Born order the Wick square contains a radial logarithm, whereas the conformally improved massless stress has no subtraction-scale logarithm and equals
−(g²)/(576π⁴α³r⁴)diag (1, 1, −3, 1).
The normalization g = λc/2 identifies this as the specified order-λ²c² contact sector, not the full quartic correction.
Second, for compact regular interaction vertices separated from the observation region, we construct all three connected order-λ² real-scalar two-point contractions by explicit Taylor extensions of the squared and cubed propagators. The resulting exterior numerical insertions have a controlled Friedrichs massless limit and local free-field tensor identities. We exhibit a regular bulk normalization freedom that cannot be classified by the three free apex-contact parameters.
The contribution is a convention-complete separation of infrared stabilization, collision renormalization, and massless continuation, with exact formulas for a nontrivial second-order apex-sensitive sector. General codimension-two coupling renormalization and interacting conical field theory are established subjects. The complete unswitched second-order stress and charged-current package, intersecting apex collisions, and a positive interacting state remain unconstructed. No finite Enter record acquires these continuum quantities, and no Einstein dynamics, backreaction, or spacetime emergence is inferred.
Bibliographic record
- Title
- I-B-VP-3-Ex1 Massive second-order insertions, defect renormalization, and controlled massless limits
- Author
- Cheung, Ngo
- Publication date
- 2026/10/18
- Institution
- Enter
- Report number
- I-B-VP-3-Ex1-v2
- Publisher
- Cheung Ngo Medical Limited
- First page
- 1
- Last page
- 39
- Language
- en